We begin by reviewing the theory of symplectic vortices introduced by
Mundet i Riera [1]} and Salamon and collaborators
[2]}. Let \(\Sigma \)  be a compact complex curve, \(G\)  a
compact Lie group, and \(\pi : P \rightarrow \Sigma \)  a smooth principal
\(G\) -bundle. Given any left \(G\) -manifold \(F\)  we have a left action of
\(G\)  on \(P \times F\)  given by \(g(p,f) = (pg^{-1},gf)\)  and we denote by
\(P(F) = (P \times F)/G\)  the quotient, that is, the associated fiber
bundle with fiber \(F\) . Let \(X\)  be a compact Hamiltonian \(G\) -manifold
with symplectic form \(\omega \)  and moment map \(\Phi :X \rightarrow \mathfrak {g}^*\) . The
action of \(G\)  on \(X\)  induces an action on \(J(X)\) ; and we denote by
\(J(X)^G\)  the invariant subspace. Let \(\psi : \Sigma \rightarrow BG\)  be a
classifying map for \(P\) , so that \(P \cong \psi ^* EG\)  and \(P(X) \cong \psi ^* EG \times _G X \cong \psi ^* X_G\)  where \(X_G = EG \times _G X\) .
Continuous sections \(u: \Sigma \rightarrow P(X)\)  are in one-to-one
correspondence with lifts of \(\psi \)  to \(X_G\) . The homology class
\(\deg (u)\)  of the section \(u\)  is defined to be the homology class
\(\deg (u) \in H_2^G(X,\mathbb {Z})\)  of the corresponding lift. Let \(\mathcal {A}(P)\)  be
the space of smooth connections on \(P\) , and \(P(\mathfrak {g})\)  the adjoint
bundle. For any \(A \in \mathcal {A}(P)\) , let \(F_A \in \Omega ^2(\Sigma ,P(\mathfrak {g}))\) 
the curvature of \(A\) . Any connection \(A \in \mathcal {A}(P)\)  induces a map of
spaces of almost complex structures
\( J(X)^G \rightarrow J(P(X)), \ \ J \mapsto J_A\) 
